Search any question & find its solution
Question:
Answered & Verified by Expert
$A=\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]$, then which of the following is/are correct?
I. $A B$ is defined
II. $B A$ is defined
III. $A B=B A$
Select the correct answer using the codes given below.
Options:
I. $A B$ is defined
II. $B A$ is defined
III. $A B=B A$
Select the correct answer using the codes given below.
Solution:
2245 Upvotes
Verified Answer
The correct answer is:
I, II and III
Let $\mathrm{A}=\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]$
Now, $\mathrm{AB}=\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]=\left[\begin{array}{ll}3 & 5 \\ 0 & 8\end{array}\right]$
and $\mathrm{BA}=\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]=\left[\begin{array}{ll}3 & 5 \\ 0 & 8\end{array}\right]$
$\Rightarrow \quad \mathrm{AB}=\mathrm{BA}$
Hence, all the three statements are correct.
Now, $\mathrm{AB}=\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]=\left[\begin{array}{ll}3 & 5 \\ 0 & 8\end{array}\right]$
and $\mathrm{BA}=\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]=\left[\begin{array}{ll}3 & 5 \\ 0 & 8\end{array}\right]$
$\Rightarrow \quad \mathrm{AB}=\mathrm{BA}$
Hence, all the three statements are correct.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.